Savings Accounts - DePaul University



Savings Accounts

Savings accounts are an example of exponential growth with a fixed percent increase. The money grows exponentially by a fixed interest rate which is called the annual percentage rate (APR).

We will look at 3 different compound situations- annual, quarterly, and monthly.

Accounts that compound annually calculate and add interest into the account once a year.

Example: deposit $500 compounded annually with an annual percentage rate (APR) of 3%

Using the formula P*(1+r), the formula in B3 is =B2*(1+.03)

|year |money |

|0 | $ 500.00 |

|1 | $ 515.00 |

|2 | $ 530.45 |

|3 | $ 546.36 |

|4 | $ 562.75 |

|5 | $ 579.64 |

|6 | $ 597.03 |

|7 | $ 614.94 |

|8 | $ 633.39 |

|9 | $ 652.39 |

|10 | $ 671.96 |

To find the value of the account after 10 years without using excel, use the “by hand” formula Y = P * (1+r)x : 500*(1+.03)^10=$671.96

For accounts compounded quarterly, you receive one fourth of the annual percentage rate each quarter. Therefore in the formula you must divide the APR by 4. Remember each period is one quarter of a year NOT one year.

Below are the first two years (8 quarters) of the account:

The formula in B3 is =B2*(1+.03/4)

|quarter |money |

|0 | $ 500.00 |

|1 | $ 503.75 |

|2 | $ 507.53 |

|3 | $ 511.33 |

|4 | $ 515.17 |

|5 | $ 519.03 |

|6 | $ 522.93 |

|7 | $ 526.85 |

|8 | $ 530.80 |

To find the amount after 2 years in one step, you need to remember to divide the APR by 4 and raise it to the number of quarters (years*4).

500*(1+.03/4)^8=$530.80.

Similarly, for compounded monthly divide the APR by 12 and for the “by hand” formula multiply the years by 12. Remember each period in the table is one month NOT one year.

The formula in B3 is =B2*(1+.03/12)

|month |money |

|0 | $ 500.00 |

|1 | $ 501.25 |

|2 | $ 502.50 |

|3 | $ 503.76 |

|4 | $ 505.02 |

|5 | $ 506.28 |

|6 | $ 507.55 |

|7 | $ 508.82 |

|8 | $ 510.09 |

|9 | $ 511.36 |

|10 | $ 512.64 |

|11 | $ 513.92 |

|12 | $ 515.21 |

|13 | $ 516.50 |

|14 | $ 517.79 |

|15 | $ 519.08 |

|16 | $ 520.38 |

|17 | $ 521.68 |

|18 | $ 522.98 |

|19 | $ 524.29 |

|20 | $ 525.60 |

|21 | $ 526.92 |

|22 | $ 528.23 |

|23 | $ 529.55 |

|24 | $ 530.88 |

So to calculate the amount in an account compounded monthly after 2 years:

500*(1+.03/12)^24=$530.88

If you compare the value after 2 years in all three accounts, you will find that the monthly account has more than the quarterly account which has more than the yearly account ($530.88 vs $ 530.80 vs $530.45). Assuming you start with the same amount and the same APR, the more often the interest is compounded, the more money you make.

What we have noticed is that with an account that compounds other than annually, the account makes more than the annual percentage rate. The actual rate earned by the account is called the Annual Percentage Yield (APY). The APY is the percent change in value of the account after ONE year.

APY = (new-old)/old = (amount after one year - original amount) / original amount

For our quarterly example, (515.17-500)/500= 3.034%

For our monthly example, (515.21-500)/500 = 3.042%

Remember to put it in percent form with a few decimal places. This shows that the account is earning an APY of 3.034%

The annual percentage yield will allow you to compare two different accounts to determine which will earn more money. For example, if you were offered an account that compounded quarterly at a 6.2% APR and another account that compounded monthly at a 6.15% APR, you don’t know which will earn you more money without calculating the APY. To do so, assume a starting value for each account ($100) and figure out how much is in each account after one year. To do this you can either set up a table in Excel or use the “by hand” formula:

|quarter |money |

|0 | $ 100.00 |

|1 | $ 101.55 |

|2 | $ 103.12 |

|3 | $ 104.72 |

|4 | $ 106.35 |

Or 100*(1+.062/4)^4 = 106.35

Then calculate the APY =(106.35-100)/100 = 6.35%

|month |money |

|0 | $ 100.00 |

|1 | $ 100.51 |

|2 | $ 101.03 |

|3 | $ 101.55 |

|4 | $ 102.07 |

|5 | $ 102.59 |

|6 | $ 103.11 |

|7 | $ 103.64 |

|8 | $ 104.17 |

|9 | $ 104.71 |

|10 | $ 105.24 |

|11 | $ 105.78 |

|12 | $ 106.33 |

Or 100*(1+.0615/12)^12 = 106.33

APY = (106.33-100)/100 = 6.33%

This shows us that the quarterly account has a higher APY and therefore earns more money.

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